Assessment of k-epsilon Turbulence Model for Compressible Flows using Direct Simulation Data

The k-epsilon turbulence model is widely used to simulate incompressible and
compressible flows. In this model, transport equations for the turbulent kinetic energy, k, and its dissipation rate, epsilon, are solved. The Reynolds stress is then modeled in terms of k and epsilon, along with a damping function to account for the low Reynolds number, Re, effects close to a solid wall. The modeling of the unclosed terms in k and epsilon equations, and the low Re damping function are mostly based on dimensional arguments. The validity of these assumptions often limit the performance of the model when applied to engineering problems. The k-epsilon turbulence model has been tested against a wide range of experimental data. However, most of the data are limited to the mean flow quantities. The higher order correlations involved in the unclosed terms are difficult to measure experimentally. By comparing the model prediction of the mean flow quantities with the experimental data, one can assess the overall performance of the turbulence model but cannot evaluate the assumptions made for each unclosed term. In this regard, a direct numerical simulation (DNS) database is very useful, wherein the unclosed terms can be evaluated exactly and compared to their modeled counterpart.

We evaluate the \keps turbulence model using DNS data of a Mach 4 boundary layer. We find that the low Reynolds number damping functions for the Reynolds stress must be corrected by the density ratio to match the DNS data. We present the budget of the k equation and assess the modeling of the various source terms. The models for all the source terms, except for the production and dilatational dissipation terms, are found to be adequate. Finally, we present the solenoidal dissipation rate equation and compute its budget using the boundary layer data. We compare this equation with the dissipation rate equation in an incompressible flow to show the equivalence between the two equations. This is the basis for modeling the solenoidal dissipation equation. However, an additional term in the equation due to variation of fluid viscosity needs to be modeled.

Budget of the turbulent kinetic energy in a Mach 4 boundary layer